Our last ingredient for Walker's strategy is the MinBox game, a variant of the Box Game [1]}, which was introduced in [2]} and motivated by [3]}. Let positive integers \(n,D,b\)  and a real \(\alpha \in (0,1)\)  be given. The MinBox\((n,D,\alpha ,b)\)  is a \((1:b)\)  Maker-Breaker game played on a
family of \(n\)  pairwise disjoint boxes \(F_1,\ldots ,F_n\) , each of size at least \(D\) , where Maker wins if and only if she occupies at least an \(\alpha \) -fraction of elements in each of the \(n\)  boxes.
To analyse the game, let us use the following definitions throughout the game: For each box \(F\) , let \(w_M(F)\)  and \(w_B(F)\)  denote the number of elements that Maker and Breaker have claimed in \(F\) , respectively, and set \(\text{dang}(F) := w_B(F) - b \cdot w_M(F)\) . Call \(F\)  free if every element of \(F\)  is not claimed yet, and call it active if \(w_M(F)<\alpha |F|\) . In [2]} the following was proven.
